Mixed-valence compounds with strong couplings between electronic states constitute one of the most challenging types of multireference systems for electronic structure theory. Previous work on a model mixed-valence compound, the 2,2′,6,6′-tetrahydro-4H,4′H-5,5′-spirobi[cyclopenta[c]pyrrole] cation, showed that multireference perturbation theory (MRPT) can give a physical energy surface for the mixed-valence compound only by going to the third order or by using a scheme involving averaging orbital energies in a way specific to mixed-valence systems. In this study, we show that second-order MRPT methods (CASPT2, MS-CASPT2, and XMS-CASPT2) can give good results by calculating the Fock operator for the zeroth-order Hamiltonian using the state-averaged density matrix. We also show that state-interaction pair-density functional theory (SI-PDFT) is free from the unphysical behavior of previously tested second-order MRPT methods for this prototype mixed-valence compound near the avoided crossing. This is very encouraging because of the much lower cost in applying SI-PDFT to large or complex systems.
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